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Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction. There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness ( completeness as a metric space ).
The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. [2] It can be used to prove many of the fundamental results of real analysis , such as the intermediate value theorem , the Bolzano–Weierstrass theorem , the extreme value theorem , and the Heine ...
Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property. The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first ...
Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences. Archimedean property : for every real number x , there is an integer n such that x < n {\displaystyle x<n} (take, n = u + 1 , {\displaystyle n=u+1,} where u {\displaystyle u} is the least upper bound of the integers less than x ).
The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness . If an ordered set S {\displaystyle S} has the property that every nonempty subset of S {\displaystyle S} having an upper bound also has a least upper ...
When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which comes from Peano's ε).
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions ...
The Dedekind–MacNeille completion is characterized among completions of S by this property. [14] The Dedekind–MacNeille completion of a Boolean algebra is a complete Boolean algebra; this result is known as the Glivenko–Stone theorem, after Valery Ivanovich Glivenko and Marshall Stone. [15]